Brownian motion ensembles and parametric correlations of the transmission eigenvalues: Application to coupled quantum billiards and to disordered wires

The parametric correlations of the transmission eigenvalues $T_i$ of a $N$-channel quantum scatterer are calculated assuming two different Brownian motion ensembles. The first one is the original ensemble introduced by Dyson and assumes an isotropic diffusion for the $S$-matrix. The second Brownian motion ensemble assumes for the transfer matrix $M$ an isotropic diffusion yielded by a multiplicative combination law. We review the qualitative differences between transmission through two weakly coupled quantum dots and through a disordered line and we discuss the mathematical analogies between the Fokker-Planck equations of the two Brownian motion models.Comment: 33 pages, 7 postscript figures, the presented abstract is shortened in comparison to the abstract of the pape

Fractional Langevin equation

We investigate fractional Brownian motion with a microscopic random-matrix model and introduce a fractional Langevin equation. We use the latter to study both sub- and superdiffusion of a free particle coupled to a fractal heat bath. We further compare fractional Brownian motion with the fractal time process. The respective mean-square displacements of these two forms of anomalous diffusion exhibit the same power-law behavior. Here we show that their lowest moments are actually all identical, except the second moment of the velocity. This provides a simple criterion which enables to distinguish these two non-Markovian processes.Comment: 4 page

Two harmonically coupled Brownian particles in random media

We study the behaviour of two Brownian particles coupled by an elastic harmonic force in a quenched disordered medium. We found that to first order in disorder strength, the relative motion weakens (with respect to the reference state of a Brownian particle with the double mass) the effect of the quenched forces on the centre of mass motion of the Brownian particles, so that the motion will become less subdiffusive (superdiffusive) for potential (solenoidal) disorder. The mean-square relative distance between the particles behaves in a different way depending of whether the particles are free to move or one particle is anchored in the space. While the effect of nonpotential disorder consists in increasing the mean-square distance in both cases, the potential disorder decreases the mean-square distance, when the particles are free to move, and increases it when one particle is anchored in the space.Comment: 8 pages, 3 figure

Brownian Motion in Robertson-Walker Space-Times from electromagnetic Vacuum Fluctuations

We consider classical particles coupled to the quantized electromagnetic field in the background of a spatially flat Robertson-Walker universe. We find that these particles typically undergo Brownian motion and acquire a non-zero mean squared velocity which depends upon the scale factor of the universe. This Brownian motion can be interpreted as due to non-cancellation of anti-correlated vacuum fluctuations in the time dependent background space-time. We consider several types of coupling to the electromagnetic field, including particles with net electric charge, a magnetic dipole moment, and electric polarizability. We also investigate several different model scale factors.Comment: 29 pages, 7 figure